Expand
$$\left( z + \frac { 1 } { z } \right) \left( z - \frac { 1 } { z } \right)$$
Hence, or otherwise, expand
$$\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }$$
Use De Moivre's theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\) then
$$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
Write down a corresponding result for \(z ^ { n } - \frac { 1 } { z ^ { n } }\).
Hence express \(\cos ^ { 4 } \theta \sin ^ { 2 } \theta\) in the form
$$A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
where \(A , B , C\) and \(D\) are rational numbers.